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<ノート>多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(13)量子統計力学(5)

邦夫 橋爪

Published Dec 1, 2009• 0 citations
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Abstract

In this paper, next subjects are discussed. §35. Magnetization and magnetic susceptibility in the canonical ensemble and in the grand canonical ensemble. §36. Landau levels. §37. Landau diamagnetism and magnetic susceptibility. §38. Quantization of area enclosed by Landau orbits in k-space, quantization of area enclosed by cyclotron orbits in real space and quantization of magnetic flux.

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In this paper, next subjects are discussed. §35. Magnetization and magnetic susceptibility in the canonical ensemble and in the grand canonical ensemble. §36. Landau levels. §37. Landau diamagnetism and magnetic susceptibility. §38. Quantization of area enclosed by Landau orbits in k-space, quantization of area enclosed by cyclotron orbits in real space and quantization of magnetic flux.
[Synopsis] 0The alignment of the electron spin with the external magnetic field give rise to paramagnetism. Whereas, the orbital motions of the electrons give rise to diamagnetism. In this paper we consider the idealized problem of a free spinless electron gas in an external magnetic field H＝H^^ｚ . A free electron moves in a circular orbital quantized normal to the uniform magnetic field with the cycrotron frequency ω＝<—eμ_0H>/m. The energy eigenvalues in two-dimesional electron system E_j ＝hω_0(j＋1/2)＝βH(2j＋1) (j＝0,1,2,･･･) are the Landau energy levels. β is a Bohr magneton. g＝<—eμ_0H>/hL^2 is the degeneracy of a Landau level . In the low-temperature limit of electron in Landau levels, the electrons have a tendency to occupy the lowest available levels. As the magnetic field is decrease, each Landau level can accommodate fewer electrons because the degeneracy g is decreased. Consequently, some electrons will be forced to jump up to a higher level. This cause the oscillation of the low-temperature magnetic susceptibility X_m as a inverse function of the magnetic field 1/H. The magnetization M also oscillates as a function of the magnetic field. We call these phenomena the de Haas-Van Alphen effect.
[Synopsis]In this paper, next subject is discussed. § 40. Magnetic properties of an imperfect electron gas. We discuss the effects of interparticle interaction affecting the magnetic properties of an electron gas. Two electrons prefer to be in an antisymmetric orbital wave function to minimize the repulsive energy. But in fermions, total wave function of two electrons with spin-1/2 must be antisymmetric. Therefore an antysimmetric orbital wave function requires a symmetric spin wave function. Thus the repulsive interaction would align the spins of the electrons. And the effect of a repulsive interaction enhances the paramagnetism. We here demonstrate this effect by considering an imperfect electron gas at very low temperature.